Projection Oracle in Convex Sampling

• Projection Oracle is an algorithmic subroutine that computes the exact Euclidean projection onto convex bodies, ensuring every sample lies within the set.
• It underpins the alternating sampling framework by centering Gaussian proposals at the projected point, thereby reducing rejection rates in high-dimensional MCMC.
• Theoretical analysis demonstrates that projection oracles achieve optimal iteration complexity with improved reliability and minimal oracle calls compared to alternatives.

A projection oracle, in the context of convex computational geometry and high-dimensional Markov chain Monte Carlo (MCMC) sampling, refers to an oracle (algorithmic subroutine) that, given any point y in the ambient space, efficiently solves the convex quadratic program $$\operatorname{proj}_K(y) = \arg\min_{x \in K} \|x-y\|^2$$. The availability of such an oracle enables exact geometric operations fundamental to numerous randomized algorithms for convex bodies, in particular the construction of rejection samplers for restricted Gaussian distributions, which underlie modern uniform samplers and proximal MCMC frameworks. The computational and theoretical tradeoffs of the projection oracle as compared to weaker separation or membership oracles have been analyzed in the context of uniform sampling on high-dimensional sets (Dang et al., 3 Oct 2025).

1. Projection Oracle in the Alternating Sampling Framework

In oracle-based uniform sampling from convex bodies (Dang et al., 3 Oct 2025), each iteration of the Alternating Sampling Framework (ASF) consists of

• a Gaussian proposal update, and
• a restricted Gaussian oracle (RGO) step that samples from ${\mathcal{N}}(y, \eta I_d)$ restricted to the convex body $K$.

When a projection oracle is available, the RGO step leverages the exact minimizer $\operatorname{proj}_K(y)$, taking as its proposal distribution a standard Gaussian centered at this projected point: $$x \sim {\mathcal{N}}(\operatorname{proj}_K(y), \eta I_d).$$ The rejection sampling procedure then accepts x with probability determined by an auxiliary function $\mathcal{P}_1(x)$, ensuring samples are drawn exactly from the restricted Gaussian ${\mathcal{N}}(y, \eta I_d)|_K$. Acceptance is certified by $x \in K$, a guarantee that follows from the convexity and uniqueness of projections.

Lemma 2.3 in (Dang et al., 3 Oct 2025) establishes that the acceptance probability is always valid (bounded above by one) and that, for step size $\eta = 1/d^2$, the expected number of rejections per RGO invocation is $M(\sqrt{2\pi e} + 1)$, where M is a "warmness" constant describing the initial distribution's closeness to the uniform distribution on $K$.

2. Complexity Under Divergence Measures

The iteration complexity of the resulting sampler is analyzed both in Rènyi divergence and $\chi^2$ divergence. Under standard geometric assumptions (e.g., $K \subset \mathcal{B}(R)$ and $\mathcal{B}(1) \subset K$), convergence to $\epsilon$-accuracy is guaranteed in

$$O\bigg( d^2\, C_{\mathrm{LSI}}\, q\, \log\left( \frac{2\log M}{\epsilon}\right)\bigg)$$

iterations for Rènyi divergence (where $C_{\mathrm{LSI}}$ is the log-Sobolev constant for $K$), with each iteration requiring one projection oracle query and an average of $M(\sqrt{2\pi e} + 1)$ candidate samples. For $\chi^2$ divergence, the complexity is

$$O\bigg( d^2\, C_{\mathrm{PI}}\, \log\left( \frac{2(M^2+1)}{\epsilon}\right)\bigg),$$

with $C_{\mathrm{PI}}$ the Poincaré constant.

These rates match, up to constants, the best available from algorithms using only membership oracles, but guarantee that every accepted sample lies in $K$ almost surely.

Divergence	Iterations (Projection Oracle)	Oracle Calls per Iteration
Rènyi $\mathcal{R}_q$	$O(d^2\,C_{\mathrm{LSI}}\,q\,\log(1/\epsilon))$	1 projection, $O(1)$ rejections
$\chi^2$	$O(d^2\,C_{\mathrm{PI}}\,\log(1/\epsilon))$	1 projection, $O(1)$ rejections

3. Comparison with Separation and Membership Oracles

In the absence of a projection oracle, the paper describes an alternative based on a separation oracle and cutting-plane methods for approximating $\operatorname{proj}_K(y)$. This approach, unlike the projection oracle method, centers the proposal distribution at an approximate minimizer $\hat{x}$ found via a separation oracle and uses a modified, log-concave, non-Gaussian proposal density: $$\nu(x) \propto \exp\left( -\frac{1}{2\eta}(\|x-\hat{x}\|^2 - 2\sqrt{2\eta/d}\|x-\hat{x}\|) \right).$$ Sampling from this proposal is reduced to a one-dimensional problem, and each RGO step now requires $O(d \log (d\gamma/\alpha))$ separation oracle calls for a precision parameter $\alpha$ and geometry-dependent constant $\gamma$.

Although both approaches share matching iteration complexity in divergence metrics, the projection oracle reduces the number of oracle calls per iteration to a single, exact, and optimal projection.

4. Uniform Sampler Robustness and Accuracy

Unlike the “in-and-out” sampler, which is purely membership-oracle-based (with a nonzero failure probability in the rejection step), the projection-oracle-based method achieves failure-free acceptance and guarantees that all samples reside in $K$. The uniformity and distributional correctness of the output follow from the properties of Euclidean projection and the acceptance probability structure established for the proposal kernel.

Moreover, the separation-oracle variant retains the same rigorous guarantees in divergence up to the precision set by the cutting-plane approximation of the projection. The use of log-Sobolev and Poincaré constants in the analysis allows extension of the results to general convex bodies (with improvement in the isotropic case).

5. Theoretical Implications and Relation to Prior Work

The equivalence in iteration complexity (up to constants) to the best membership-oracle-based algorithms—such as the In-and-Out sampler—demonstrates the theoretical optimality of projection-oracle uniform samplers for convex bodies, with strict improvement in per-iteration reliability and required oracle strength. The iteration bounds for convergence in Rènyi and $\chi^2$ divergence highlight the fundamental role of log-Sobolev and Poincaré inequalities in quantifying sample complexity under exact or approximate projection oracles.

Earlier work relying on membership oracles requires careful handling of sampling failures and often introduces restart procedures to guarantee unbiasedness, while projection-oracle-based methods avoid this overhead entirely via the uniqueness and computability of the projected minimizer.

6. Practical Considerations and Implementation

When a projection oracle is available (e.g., for polyhedra, balls, ellipsoids, or certain spectral sets), each MCMC iteration of the framework requires a single projection and an average number of O(1) rejection samples. This property provides favorable scaling in high dimensions and underlies the practical deployability of the approach. When only separation oracles are available, the additional complexity per iteration is controlled and explicit, and the bottleneck shifts to the precision of the cutting-plane solver.

The projection oracle thus forms a foundational primitive in geometric MCMC, enabling rigorous uniform sampling with optimal per-iteration accuracy guarantees in both theory and practice. The results in (Dang et al., 3 Oct 2025) systematically establish its tradeoffs and performance alongside alternative oracles.